Uncertain Fractional Order Chaotic Systems Tracking Design via Adaptive Hybrid Fuzzy Sliding Mode Control
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1 Int. J. of Computers, Communications & Control, ISSN , E-ISSN Vol. VI (011, No. 3 (September, pp Uncertain Fractional Orer Chaotic Systems Tracking Design via Aaptive Hybri Fuzzy Sliing Moe Control T.C. Lin, C.H. Kuo, V.E. Balas Tsung-Chih Lin Feng-Chia University, 4074, Taichung, Taiwan tclin@fcu.eu.tw Chia-Hao Kuo Ph.D Program in Electrical an Communications Engineering Feng-Chia University, Taichung, Taiwan peterqo0@hotmail.com Valentina E. Balas Aurel Vlaicu University of Ara, Romania B-ul Revolutiei 77, Ara, Romania balas@rbalas.ro Abstract: In this paper, in orer to achieve tracking performance of uncertain fractional orer chaotic systems an aaptive hybri fuzzy controller is propose. During the esign proceure, a hybri learning algorithm combining sliing moe control an Lyapunov stability criterion is aopte to tune the free parameters on line by output feeback control law an aaptive law. A weighting factor, which can be ajuste by the trae-off between plant knowlege an control knowlege, is aopte to sum together the control efforts from inirect aaptive fuzzy controller an irect aaptive fuzzy controller. To confirm effectiveness of the propose control scheme, the fractional orer chaotic response system is fully illustrate to track the trajectory generate from the fractional orer chaotic rive system. The numerical results show that tracking error an control effort can be mae smaller an the propose hybri intelligent control structure is more flexible uring the esign process. Keywors: Fractional orer chaotic systems; fuzzy logic control, aaptive hybri control. 1 Introuction in omain Due mainly to its emonstrate applications in numerous seemingly iverse an wiesprea fiels of science an engineering, fractional calculus has gaine consierable popularity an importance uring past three ecaes 1-. In control system, ue to the fact that the theoretical aspects are well establishe, fractional orer controllers are successfully use to enhance the performance of the feeback control loop. It is observe that the escription of some systems is more accurate when the fractional erivative is use. Nowaays, many fractional-orer ifferential systems behave chaotically, such as the fractional-orer Chua s system 3, the fractionalorer Duffing system 4, the fractional-orer system, the fractional-orer Chen s system 5, the fractional-orer cellular neural network 6, the fractional-orer neural network 7. The tracking problem of fractional orer chaotic systems is first investigate by Deng an Li 1 who carrie out tracking in case of the two fractional Lü systems. Afterwars, they stuie chaos tracking of the Chen system with a fractional orer in a ifferent manner -4. Base on the universal approximation theorem, 9-0 (fuzzy logic controllers are general enough to perform any nonlinear control actions there is rapily growing interest in systematic Copyright c by CCC Publications
2 Uncertain Fractional Orer Chaotic Systems Tracking Design via Aaptive Hybri Fuzzy Sliing Moe Control 419 esign methoologies for a class of nonlinear systems using fuzzy aaptive control schemes. Like the conventional aaptive control, the aaptive fuzzy control is classifie into irect an inirect fuzzy aaptive control categories 9, A irect aaptive fuzzy controller uses fuzzy logic systems as controller in which linguistic fuzzy control rules can be irectly incorporate into the controller. On the other han, an inirect aaptive fuzzy controller uses fuzzy escriptions to moel the plant in which fuzzy IF-THEN rules escribing the plant can be irectly incorporate into the inirect fuzzy controller. Moreover, a hybri aaptive fuzzy controller can be constructe using a weighting factor to sum together the control efforts from inirect aaptive fuzzy controller an irect aaptive fuzzy controller. Although the concept of sliing moe control (SMC an the theory of fractional orer system are well known, their integration, fractional sliing moe control, is an interesting file of research welt on this paper with some applications 8. The motivation of this paper stans on two riving forces: One, most systems in the reality isplay behavior characterize best in time omain of fractional operators, the other, the uncertainties on the process ynamics can appropriately be alleviate by utilizing SMC technique. In this paper, by combining the approximate mathematical moel, linguistic moel escription an linguistic control rules into a single aaptive fuzzy controller, an aaptive hybri fuzzy controller is propose to achieve prescribe tracking performance of fractional orer chaotic systems. A new aaptive hybri fuzzy SMC algorithm incorporate Lyapunov stability criterion is propose so that not only the stability of aaptive fuzzy control system is guarantee but also the influence of the approximation error an external isturbance on the tracking error can be attenuate to an arbitrarily prescribe level. This paper is organize as follows: In section, an introuction to fractional erivative an its relation to the approximation solution will be aresse. Section 3 generally proposes aaptive hybri fuzzy SMC of uncertain fractional orer systems in presence of uncertainty an its stability analysis. In Section 4, application of the propose metho on fractional orer expression chaotic system is investigate. Finally, the simulation results an conclusion will be presente in Section 5. Basic efinition an preliminaries for fractional orer systems The concept of fractional calculus is popularly believe to have steame from a question raise in the year 1695 by Marquis e L Hoptial to Gottfrie Wilhelm Leibniz. It is a generalization of integration an ifferentiation to non-integer orer funamental operator, enote by a D q t, where a an t are the limits of the operator. This operator is a notation for taking both the fractional integral an functional erivative in a single expression efine as 1 q ad q t, q > 0 q t = 1 q = 0 a t (τ q, q < 0 There are some basic efinitions for the general fractional an the commonly use efinitions are Grunwal-Letnikov an Riemann-Liouville 1. The Grunwal-Letnikov efinition is expresse as (1 ad q t f(t = lim h 0 t a h ( a ( 1 j b j=0 f(t jh (
3 40 T.C. Lin, C.H. Kuo, V.E. Balas where. is the integer part. The simplest an easiest efinition is Riemann-Liouville efinition given as n ad q t f(t = 1 Γ(n q t n t 0 f(τ τ (3 (t τ q n+1 where n is the first integer which is not less q, i.e., n 1 < q < n, an Γ is the Gamma function. The numerical simulation of a fractional ifferential equation is not simple as that of an orinary ifferential equation. In this paper, the algorithm which is an improve version of Aams-Bashforth-Moulton algorithm to fin an approximation for fractional orer systems base on preictor-correctors is given. Consier the following ifferential equation where ad q t y(t = r(y(t, t, 0 ad q t y(t = t T an y(k (0 = y (k o, k = 0, 1,,..., m 1 (4 1 t Γ(m q 0 m t y(t, m an m is the first integer larger the q. Volterra integral equation 1 escribe as y(t = q 1 k=0 y (k t k 0 k! + 1 Γ(q f (m (τ (t τ q m+1 τ, m 1 < q < m q = m (5 The solution of the equation (4 is equivalent to t 0 (t λ q 1 r(y(λ, λλ (6 Let h=t/n, t n = nh, n=0,1,, N. Then (6 can be iscretize as follows. y h (t n+1 = q 1 k=0 y (k t k n k! h q Γ(q + r(yp h (t n+1, t n+1 + where preict value y p h (t n+1 is etermine by h q Γ(q + n a j,n+1 r(y h (t j, t j (7 j=0 an y p h (t n+1 = q 1 k=0 y (k t k n hq k! Γ(q n b j,n+1 r(y h (t j, t j (8 n q+1 (n q(n + 1 q, j = 0 a j,n+1 = (n j + q+1 + (n j q+1 (n j + 1 q+1 1 j n 1 j = n + 1 j=0 (9 The approximation error is given as b j,n+1 = hq q ((n + 1 jq (n j q (10 max y(t j y h (t j = O(h p (11 j=0,1,, N where p=min(,1+q. Therefore, the numerical solution of a fraction orer chaotic system iscusse in this paper can be obtaine by applying the above mentione algorithm.
4 Uncertain Fractional Orer Chaotic Systems Tracking Design via Aaptive Hybri Fuzzy Sliing Moe Control 41 3 Aaptive hybri fuzzy sliing moe control of uncertain fractional orer chaotic systems In this section, we stuy aaptive hybri fuzzy tracking control of uncertain fractional orer chaotic systems, i.e., to force output trajectory which is obtaine by the algorithm mentione in section of the response system to track output trajectory of the rive system. Consier a fractional orer chaotic ynamic system x (nq = f(x, t + g(x, tu + (t, y = x 1 (1 where x = x 1, x,..., x n T = x, x (q, x (q,..., x ((n 1q T is the state vector, f(x, t an g(x, t are unknown but boune nonlinear functions which express system ynamics, (t is the external boune isturbance, (t D, an u(t is the control input. The control objective is to force the system output y to follow a boune reference signal y which is the output trajectory of a rive system, uner the constraint that all signals involve must be boune. To begin with, the reference signal vector y an the tracking error vector e will be efine as e = y x = y = y, y (q T,..., y((n 1q R n, e, e (q, e (q,..., e ((n 1q T R n, e (iq = y (iq x (iq = y (iq y (iq In general, in the space of the error state a sliing surface is efine by ( s(x, t = (ke = k 1 e + k e (q k n 1 e (n q + e (n 1q (13 where k = k 1, k,..., k n 1, 1 in which the k i s are all real an are chosen such that h(r = n k ir (i 1q, k n = 1 is a Hurwitz polynomial where r is a Laplace operator. The tracking problem will be consiere as the state error vector e remaining on the sliing surface s(x, t = 0 for all t 0. The sliing moe control process can be classifie into two phases, the approaching phase with s(x, t 0 an the sliing phase with s(x,t= 0 for initial error e(0 = 0. In orer to guarantee that the trajectory of the state error vector e will translate from the approaching phase to the sliing phase, the sufficient conition s(x, tṡ(x, t η > 0 (14 must be satisfie. Two type of control law must be erive separately for those two phases escribe above. In the sliing phase, it implies s(x, t = 0 an s (q (x, t = 0. In orer to force the system ynamics to stay on the sliing surface, the equivalent control u can be erive as follows: If f(x, t an g(x, t are known an free of external isturbance, i.e., (t=0, taking the erivative of the sliing surface with respective to time, we get = ( n 1 s (q = ( n 1 k i e (iq f(x,t g(x,t u eq c i e (iq + e (nq = y (n ( n 1 k i e (iq + y (nq y(nq n 1 = k i e (i + f(x + b(xu(t x (n = 0 (15 Therefore, the equivalent control can be obtaine as
5 4 T.C. Lin, C.H. Kuo, V.E. Balas ( u = 1 n 1 k i e (iq f(x, t + y (nq g(x, t On the contrary, in the approaching phase, s(x, t 0, an approaching-type control u ap must be ae in orer satisfy the sufficient conition (4 an the complete sliing moe control will be expresse as (16 u = u u ap, u ap = ψ h sgn(s (17 where ψ h η > 0. To obtain the sliing moe control (17, the system functions f(x, t, g(x, t an switching parameter ψ h must be known in avance. However, f(x, t an g(x, t are unknown an external isturbance, (t 0, the ieal control effort (16 cannot be implemente. We replace f(x, t, g(x, t an u ap by the fuzzy logic system f(x θ f, g(y θ g an h(s θ h in specifie form as 9, 17-19, i.e., f(x θ f = ξ T (xθ f, g(x θ g = ξ T (xθ g, h(s θ h = T (sθ h (18 let h(s θ h = D + ψ h + ω max when s(x, t is outsie the bounary layer. Here the fuzzy basis functions ξ(x an (s epen on the fuzzy membership functions an is suppose to be fixe, while θ f, θ g an θ h are ajuste by aaptive laws base on Lyapunov stability criterion. Therefore, epening on plant knowlege an control knowlege, a hybri aaptive fuzzy controller can be constructe by incorporating both fuzzy escription an fuzzy control rules using a weighting factor α to combine the inirect aaptive fuzzy controller an the irect aaptive fuzzy controller. Base on the trae-off between plant knowlege an control knowlege, the weighting factor α 1, 1 can be ajuste. Therefore, the total control effort can be expresse as u c = αu i + (1 αu (19 where the irect aaptive fuzzy controller u an the inirect aaptive fuzzy controller u i are given as follows: u (x = u D (x θ D h(s θ h g(x, t an u i(x = n 1 1 g(x θ g where u D (x θ is obtaine by fuzzy logic system specifie as k i e (iq + y (nq f(x, θ f h(s θ h (0 u D (x θ D = ξ T (xθ D (1 The optimal parameter estimations θ f, θ g, θ h an θ D are efine as θ f = arg min θ f Ω f sup f(x θ f f(x, t, θ g = arg min θg Ω g sup g(x θ g g(x, t x Ω x x Ω x θ D = arg min θd Ω D sup u D (x θ g u x Ω x, θ h = arg min θ h Ω h sup h(s θ h u ap x Ω x
6 Uncertain Fractional Orer Chaotic Systems Tracking Design via Aaptive Hybri Fuzzy Sliing Moe Control 43 where Ω f, Ω g, Ω D an Ω x are constraint sets of suitable bouns on θ f, θ g, θ h, θ D an x respectively an they are efine as Ω f = {θ f θ f M f }, Ω g = {θ g θ g M g }, Ω D = {θ D θ D M D }, Ω h = {θ h θ h M h } an Ω x = {x x M x }, where M f, M g, M D, M h an are positive constants. By using (0, (1, sliing surface equation (15 can be rewritten as s (q = ω + α f(x θ f f(x θ f + α g(x θ g g(x θ g u i (1 αh(s θ h αh(s θ h (1 αg(x u D (x θ D u D (x θ D + αh(s θ h αh(s θ h ( +(1 αh(s θ h (1 αh(s θ h + (t where the minimum approximation errors is efine as ω = α f(x f(x θ f + α g(x g(x θ g u i + (1 α u D (x θ u D (3 If θ f = θ f θ f, θ g = θ g θ g an, θ D = θ D θ D, we have s (q = (1 αh(s θ h + ω α θ T h (s α θ T f ξ(x α θ T g ξ(xu i +(1 αg(x θ T Dξ(x αh(s θ h (1 α θ T h + (t (4 Following the proceeing consieration, the following theorem can be obtaine. Theorem: Consier the fractional orer SISO nonlinear chaotic system (1 with control input (19, if the fuzzy-base aaptive laws are chosen as θ (q f = r 1 sξ(x, θ g (q = r sξ(xu i, θ (q D = r 3s (s an θ (q h = r 4sg(xξ(x (5 where r i > 0, i = 1 4. Then, the overall aaptive scheme guarantees the global stability of the resulting close-loop system in the sense that all signals involve are uniformly boune an the tracking error will converge to zero asymptotically. Proof: In orer to analyze the close-loop stability, the Lyapunov function caniate is chosen as V = 1 s + α θt r θ f f + α θt 1 r θ g g + α θt r θ (1 α (1 α h h + θt 4 r D θ D θt 3 r θ h h (6 4 Taking the erivative of the (6 with respect to time, we get V (q = ss (q + α (q θt f θ f r 1 + α (q θt g θ g r + α (q (1 α θt h θ h + r 4 r 3 θt θq + (1 α r 4 (q θt h θ h = (1 αsh (s θ h + sω αs θ T h (s αs θ T h ξ(x αs θ T g ξ(xu i + (1 αsg(x θ T ξ(x ash (s θ h (1 αs θ T h + s(t + α (q θt f θ f r 1 α ( θ(q θt f f r 1 sξ(x + α ( θ(q θt g r 1 r + α (q θt g θ g r g r sξ(xu i + α (q (1 α θt h θ h + r ( θ(q θt h h r 4 s (s r 4 r 3 θt θq + (1 α r 4 (q θt h θ h αs(d + ηsgn(s + 1 α ( θ(q θt + r3 sg(xξ(x (1 αs(d + ψ h sgn(s + s(t + sω (7 r 3 From the robust compensator u a an the fuzzy-base aaptive laws are given (5, after simple manipulation, we have V (q sω sψ h sgn(s = sω s ψ h (8 Using the corollary of Barbalat s Lemma 16-19, we have lim t s(x, t = 0. Therefore, lim t e(t = 0. The proof is complete.
7 44 T.C. Lin, C.H. Kuo, V.E. Balas 4 Simulation example In this section, we will apply our aaptive hybri fuzzy sliing moe controller to force the fractional orer chaotic gyro response system to track the trajectory of the fractional orer chaotic gyro rive system. Example: The fractional orer chaotic gyro rive an response systems are given as follows: Drive System: { y (q 1 = y y (q = 100 ( y y y 0.05y 3 + sin(y sin(ty 1 x (t Response System: { x (q 1 = x x (q = 100 ( x x x 0.08x 3 + sin(x sin(tx 1 x f(x 1, x + (t + u c (t where structure uncertainty f(x 1, x = 0.1sin(x 1 an external isturbance (t = 0.cos(πt. The main objective is to control the trajectories of the response system to track the reference trajectories obtaine from the rive system. The initial conitions of rive an response systems are chosen as y 1 (0, y (0 T = 1, 1 T an x 1 (0, x (0 T = 1.6, 0.8 T, respectively. For q=0.95, α = 0.7 an all esign constants are specifie as k 1 = k = 1, r 1 = 150, r = 0, r 3 = 1, r 4 = 1 an step size h = The phase portrait of the rive an response systems for free of control input is given in Figure 1. It is obvious that the tracking performance is ba without control effort supplie to response system. Figure 1: Phase portrait of chaotic rive an response systems as The membership functions for x i, are selecte as follows: µ F i(x 1 i = exp 0.5 ( x i 4, µ F i(x i = exp 0.5 ( x i.7, µ F i(x 3 i = exp 0.5 ( x i 1., µ F i(x 4 i = exp 0.5 ( x i, µ F i(x 5 i = exp 0.5 ( x i +1., µ F i(x 6 i = exp 0.5 ( x i +.7, µ F i(x 7 i = exp 0.5 ( x i +4, From the aaptive laws (5-(8, the control effort of the response system can be obtaine u c = αu i + (1 αu (9 Figure shows the trajectories of the states x i, y i an x, y, respectively. Control effort trajectory is given in Figure 3 an phase portrait, tracking performance, of the rive an response
8 Uncertain Fractional Orer Chaotic Systems Tracking Design via Aaptive Hybri Fuzzy Sliing Moe Control 45 Figure : The trajectories of the states x i, y i an x, y Figure 3: Trajectory of the control effort Figure 4: Phase portrait, tracking performance, of the rive an response systems systems is shown in Figure 4. Trajectory of the sliing surface is given is Figure 5. The maximum value of V (q (t is e-4 which is always negative efine an consequently is stable. In orer to show the robustness of the propose aaptive hybri fuzzy sliing moe control, the control effort is activate at 5 secon. The phase portrait, tracking performance, of the rive an response systems is given in Figure 6. Figure 7 shows the trajectories of the states x i, y i an x, y respectively. We can see that a fast tracking of rive an response is achieve as the control effort is activate. Control effort trajectory is given in Figure 8. Trajectory of the sliing surface is given is Figure 9. The maximum value of V (q (t is -1.73e-4 which is always negative efine an consequently is stable. Figure 5: Trajectory of the sliing surface Figure 6: Phase portrait, tracking performance, of the rive an response systems
9 46 T.C. Lin, C.H. Kuo, V.E. Balas Figure 7: The trajectories of the states x i, y i an x, y Figure 8: Trajectory of the control effort Figure 9: Trajectory of the sliing surface 5 Conclusions A novel aaptive hybri fuzzy sliing moe controller is propose to achieve tracking performance of fractional orer chaotic systems in this paper. It is a flexible esign methoology by the trae-off between plant knowlege an control knowlege using a weighting factor? aopte to sum together the control effort from inirect aaptive fuzzy controller an irect aaptive fuzzy controller. Base on the Lyapunov synthesis approach, free parameters of the aaptive fuzzy controller can be tune on line by output feeback control law an aaptive laws. The simulation example, the output trajectory of the fractional orer chaotic response system to tracking the trajectory of the fractional orer chaotic rive system, is given to emonstrate the effectiveness of the propose methoology. Bibliography 1 M. S. Tavazoei, M. Haeri, Synchronization of chaotic fractional-orer systems via active sliing moe controller, Physica A, Vol. 387, pp , 008. A.A. Kilbas, H.M. Srivastava an J.J. Trujillo, Theory an applications of fractional ifferential equations, North-Hollan Math. Stuies 04, Elsevier, Amsteram, I. Petras, A note on the fractional-orer Chua s system, Chaos, Solitons & Fractals, 38 (1, pp , X. Gao an J. Yu, Chaos in the fractional orer perioically force complex Duffing s oscillators, Chaos, Solitons & Fractals 6, pp , J.G. Lu an G. Chen, A note on the fractional-orer Chen system, Chaos, Solitons & Fractals 7, pp , 006.
10 Uncertain Fractional Orer Chaotic Systems Tracking Design via Aaptive Hybri Fuzzy Sliing Moe Control 47 6 P. Arena an R. Caponetto, Bifurcation an chaos in non-integer orer cellular neural networks, Int J Bifurcat Chaos 8 (7, pp , Petras I., A note on the fractional-orer cellular neural networks. In: Proceeings of the IEEE worl congress on computational intelligence, international joint conference on neural networks, Vancouver, Canaa; pp.16-1, S. H. Hosseninnia, R. Ghaeri, A. Ranjbar N., M. Mahmouian, S. Momani, Sliing moe synchronization of an uncertain fractional orer chaotic system, Journal Computers & Mathematics with Applications, Vol. 59, pp , L.A. Zaeh, Fuzzy logic, neural networks an soft computing. Commun. ACM 37 3, pp , X. Z. Zhang; Y.N. Wang; X. F. Yuan, H? Robust T-S Fuzzy Design for Uncertain Nonlinear Systems with State Delays Base on Sliing Moe Control, International Journal of Computers Communications & Control, 5(4:59-60, Balas, M.M., Balas, V.E., Worl Knowlege for Control Applications by Fuzzy-Interpolative Systems, International Journal of Computers Communications an Control, Vol. 3, Supplement: Suppl. S, pp. 8-3, L. X. Wang, an J. M. Menel, Fuzzy basis function, universal approximation, an orthogonal least square learning, IEEE Trans. Neural Networks, vol. 3, no. 5, pp , L. X. Wang, Aaptive Fuzzy Systems an Control: Design an Stability Analysis. Englewoo Cliffs, NJ: Prentice-Hall, K. Diethelm, An algorithm for the numerical solution of ifferential equations of fractional orer, Elec. Trans. Numer. Anal. 5, pp. 1-6, J. L. Castro, "Fuzzy logical controllers are universal approximators," IEEE Trans. Syst., Man, Cybern., 5, pp , S. S. Ge, T. H. Lee, C. J. Harris, Aaptive Neural Network Control of Robotic Manipulators, Worl Scientific Publishing Co., Singapore, C. H. Wang, T. C. Lin, T. T.. Lee an H. L. Liu, "Aaptive Hybri Intelligent Control for Unknown Nonlinear Dynamical Systems", IEEE Transaction on Systems, Man, an Cybernetics Part B, 3 (5, October, pp , C. H. Wang, H. L. Liu an T. C. Lin, "Direct Aaptive Fuzzy-Neural Control with Observer an Supervisory Control for Unknown Nonlinear Systems", IEEE Transaction on Fuzzy Systems, 10(1, pp , T. C. Lin, C. H. Wang an H. L. Liu, "Observer-base Inirect Aaptive Fuzzy-Neural Tracking Control for Nonlinear SISO Systems Using VSS an ", Fuzzy Sets an Systems, 143, pp.11-3, L. A. Zaeh, Knowlege Representation in Fuzzy Logic, IEEE Trans. Knowlege an Data Engineering, 1 (1, pp , 1989.
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